Group theoretical analysis of the vortex lattice states in the attractive Hubbard model

This paper describes a group theoretical classification of superconducting states (vortex lattice states) in a unifom magnetic field of the extended Hubbard model with on-site attraction (U < 0), or nearest-neighbor attraction (V < 0) on a two-dimensional square lattice. Using symmetries of the magnetic translation and the tetragonal rotation of the system, we obtain invariance groups of vortex lattice states. When the magnetic flux phi through a unit cell of the crystal lattice is phi(0)/p(2) (phi(0) = ch/2e is the flux quantum, p is an integer) we obtained three types of symmorphic tetragonal vortex lattice states, in which the center of point rotations is at a lattice site of the crystal lattice, corresponding to the irreducible representation of the tetragonal symmetry Dq. For these three states, locations and winding numbers of vortex lines are determined by the symmetries of their invariance group. From these three symmorphic states we derived three (two) types of non-symmorphic vortex lattice states in which the center of point rotations is at the center of a plaquette (link) of the crystal lattice. We show that d-wave, s-wave and p-wave local order parameters inevitably coexist due to the invariance magnetic translation group L of the vortex lattice state. From the irreducible projective representations (two-dimensional) of L, we show that all energy bands of the quasiparticles are at least doubly degenerate.